Inverse polynomial mappings and interpolation on several intervals

Accepted Manuscript Inverse polynomial mappings and interpolation on several intervals

A. Kroó, J. Szabados

PII: DOI: Reference:

S0022-247X(15)01165-8 http://dx.doi.org/10.1016/j.jmaa.2015.12.032 YJMAA 20061

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

2 September 2015

Please cite this article in press as: A. Kroó, J. Szabados, Inverse polynomial mappings and interpolation on several intervals, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2015.12.032

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Inverse polynomial mappings and interpolation on several intervals A. Kro´o∗ Alfr´ed R´enyi Institute of Mathematics, P.O.B. 127, H-1364 Budapest, Hungary, and Budapest University of Technology and Economics, Department of Analysis, 1111 Budapest, M˝ uegyetem rkp. 3-9, Hungary e-mail: [email protected] J. Szabados Alfr´ed R´enyi Institute of Mathematics, P.O.B. 127, H-1364 Budapest, Hungary e-mail: [email protected] December 21, 2015

Abstract In the present paper we will use the inverse polynomial image method in order to construct optimal nodes of interpolation on unions of disjoint intervals. We will show how this method works on those disjoint intervals which possess so called T-polynomials, and also prove that the method becomes ineffective in the absence of T-polynomials.

Keywords and phrases: Lagrange interpolation, Lebesgue constant and Lebesgue function, Chebyshev and T-polynomials, inverse polynomial images. Mathematical Subject Classification: 41A05. ∗

Research of both authors supported by OTKA Grant No. K111742.

1

1

Introduction

Lagrange interpolation has been a widely studied classical area of analysis for more than a century. There is a vast literature discussing the optimal choice of nodes for Lagrange interpolation on a single interval. The question of finding good nodes of interpolation on unions of disjoint intervals turned out to be a much harder problem. In the recent paper [5] the authors verified the existence of optimal nodes with Lebesgue constants of order O(log n) for any pair of intervals, but this was accomplished without providing an explicit construction. In addition, for the case of two symmetric intervals and certain pairs of non symmetric intervals explicit nodes with order O(log n) Lebesgue constants were also found. These nodes were constructed by taking inverse quadratic and cubic polynomial images of the classical Chebyshev nodes. The inverse polynomial image method was introduced by Peherstorfer [7] and Totik [10], and was successfully applied for extending various classical polynomial results from an interval to more general domains. In the present paper we shall develop a unified approach to constructing optimal nodes of interpolation on unions of disjoint intervals using the inverse polynomial image method. We will show how this method works on those intervals which possess so called T-polynomials, and also prove that the method becomes ineffective in the absence of T-polynomials. For any s ≥ 1 let −1 = a1 < b1 < · · · < as < bs = 1 be a finite partition of the interval [−1, 1], and let Js :=

s 

[ai , bi ]

(1)

i=1

be the corresponding set of s pairwise disjoint intervals. The Lebesgue function of interpolation on Js for the system of nodes Xn = {(−1 ≤)xn < xn−1 < · · · < x1 (≤ 1)} ⊂ Js , is defined as λ(Xn , x) :=

n  k=1

2

|k (x)| ,

(2)

(3)

where ωn (x) k (x) :=  , ωn (xk )(x − xk )

ωn (x) :=

n 

(x − xk ) .

k=1

Furthermore, the Lebesgue constant (the norm of the Lagrange operator) is given by λ(Xn ) := λ(Xn , x)Js , (4) where  · K is the supremum norm of the function on any compact set K. By the classical result of Faber λ(Xn ) ≥ C log n for any system of nodes on [−1, 1], while various systems of nodes on the single interval [−1, 1] are known for which the optimal order λ(Xn ) = O(log n) is attained. In [5], Theorem 1, it was shown that for any set of nodes Xn ⊂ Js λ(Xn ) ≥ C log n with some C > 0, i.e. the classic result of Faber (the case s = 1) holds for any set of disjoint intervals. (In fact, the proof of Theorem 1 in [5] can be easily extended for any compact set of positive Lebesgue measure on the real line.) On the other hand it was verified in [5], Theorem 2 that there exist systems of nodes in Js for which the Lebesgue constant is of optimal order O(log n). However, the proof of the latter result is not constructive. In addition, for certain special pairs of two disjoint intervals explicit construction of optimal nodes of interpolation was also given. In the present paper we will use the inverse polynomial image method for explicit construction of sets of nodes with optimal O(log n) order of Lebesgue constant. It turned out that a crucial role in these considerations is played by the so called T-polynomials studied in detail by Franz Peherstorfer [6]. Let us give the corresponding definition. Denote by Πm the set of algebraic polynomials of degree at most m. Now recall that a polynomial pm ∈ Πm is called the Chebyshev polynomial on Js if pm Js = 1 and its leading coefficient is maximal among all polynomials of degree at most m having norm 1 on Js . This polynomial is known to be unique. Clearly, Js ⊂ p−1 m ([−1, 1]) where p−1 m (K) := {x ∈ R : pm (x) ∈ K} denotes the real inverse polynomial image of the set K ⊂ R. The characteristic property of the Chebyshev polynomial pm is the fact that it attains values 1, −1 with alternating signs at m + 1 distinct consecutive points in Js . 3

That is there exist points zi ∈ Js , 1 ≤ i ≤ m + 1, z1 < z2 < · · · < zm+1 such that pm (zi ) = (−1)i+m+1 , 1 ≤ i ≤ m + 1. (5) Now following [6] and [8] we shall say that the Chebyshev polynomial pm is the T-polynomial on Js if |pm (x)| = 1 at m+s points of Js . Whenever pm is the T-polynomial on Js it follows that Js = p−1 m ([−1, 1]) and for each |y| < 1 the equation pm (x) = y has exactly m solutions inside Js , see [8], Lemma 1.2. In addition, it is also known that pm has m − s extremal points inside Js where |pm (x)| = 1 and all 2s boundary points of Js are also extremal. Clearly, this yields that s ≤ m. While Chebyshev polynomials exist for every compact set in R, the existence of T-polynomials of given degree m depends on the structure of Js and m. For example, in the simplest case s = m = 2, T-polynomial exists only if the pair of intervals is symmetric. When s = 2, m = 3, the set (b − a)2 J2 = [−1, a]∪[b, 1] possesses a T-polynomial if and only if a+b = 1− . 4 This statement and explicit characterization of those pairs of intervals which possess T-polynomials of degrees 2 ≤ m ≤ 6 can be found in [6]. It is also shown in [6] that for each b ∈ (−1, 1) there exists a unique a ∈ (−1, b) and a T-polynomial pm on J 2 with r extremal points in (−1, a) if and only  (m − r − 1)π if b ∈ cos , 1 . For instance, if m = 3 and r = 1, then m (m − r − 1)π = 1/2 and for each b ∈ (1/2, 1) there exists a unique cos m a ∈ (−1, b) such that the existence of the cubic T-polynomial p3 is guaranteed on J2 . T-polynomials were thoroughly investigated by F. Peherstorfer and K. Schiefermayr (see [8]) in case when s ≥ 2. There is also an alternative way of describing those systems of disjoint intervals which possess T-polynomials. One can choose an arbitrary polynomial pm ∈ Πm with m distinct real zeros and such that all its local extremal values are ≥ 1 in absolute value. Then setting Js := p−1 m ([−1, 1]) we shall obtain a system of s(≤ m) disjoint intervals for which (after rescaling to [−1, 1]), pm will be the corresponding T-polynomial. One should also note that even though not all systems of disjoint intervals possess T-polynomials it is known that any system of intervals can be approximated up to any degree of precision by systems of intervals having T-polynomials, see e.g., Peherstorfer [7] or Totik [11]. We shall explore this fact in Theorem 4 below in order to show that for any system of disjoint in4

tervals there exist convergent interpolatory processes of order (1 + )n based on n points. This extends an earlier result of Erd˝os et.al. [1] from a single interval to several intervals.

2

New results

Let pm be a T-polynomial of degree m on Js , where m ≥ s ≥ 2. For an arbitrary system of nodes Yn := {(−1 <)yn < yn−1 < · · · < y1 (< 1)},

ωn (y) :=

n 

(y − yk )

(6)

k=1

set

Xmn := p−1 m (Yn ) = {(−1 <)xmn < xmn−1 < · · · < x1 (< 1)}

(7)

be the inverse image of Yn with respect to pm . Note that since the equations pm (x) = yk have exactly m solutions in the interior of Js for each k = 1, . . . , n the system of nodes Xmn contains exactly mn points and all of them are in Js . Theorem 1. Assume that Js given by (1) possesses a T-polynomial pm of degree m, 2 ≤ s ≤ m. Then for any system of nodes (6) and Xmn = p−1 m (Yn ) we have    λ(Yn , 1) λ(Yn , −1) , λ(Xmn , x) ≤ C λ(Yn , pm (x)) + |ωn (pm (x))| + ωn (1) |ωn (−1)| (8) where x ∈ Js and C > 0 depends only on m and Js . Moreover, in the special case m = s we have λ(Xmn , x) ≤ Cλ(Yn , pm (x)),

x ∈ Js .

(9)

The above theorem provides an estimate for the Lebesgue function of any system of nodes derived by the inverse polynomial image in the presence of T-polynomials. It leads to a nice estimate for the Lebesgue constant in case when the system of nodes (6) is admissible in the sense that ωn [−1,1] = O(1), n ∈ N. |ωn (±1)|

5

(10)

Corollary 1. Let Js given by (1) possess a T-polynomial pm of degree m, 2 ≤ s ≤ m. Then for any admissible system of nodes satisfying (10) and Xmn = p−1 m (Yn ) we have λ(Xmn ) ≤ Cλ(Yn ) (11) where C > 0 depends only on m and Js . The admissibility condition (10) essentially means that at the endpoints ±1 the fundamental polynomial ωn (y) attains values of order ωn [−1,1] . Clearly, when s = m this condition is not needed at all for (11) to hold. The following proposition gives some insight to the relevance of condition (10) if s < m. We will show that if s < m λ(Xmn ) can get very large compared to λ(Yn ) whenever y1 and yn are very close to the endpoints of [−1, 1]. Proposition 1. Let pm be a T-polynomial of degree m > s on Js . Then for any system of nodes (6) and associated system Xmn = p−1 m (Yn ), we have with some C > 0 depending only on m and Js 1 1 λ(Xmn ) ≥ C min{ √ + λ(Yn , 1), √ + λ(Yn , −1)} . n 1 − y1 n 1 + yn

(12)

Example 1. Let us examine the special case of ultraspherical Jacobi polynomials, i.e. when Yn consists of zeros of the polynomials ωn (y) = Pn(α,α) (y), α > −1. It is well known that  log n, if − 1 < α ≤ −1/2, λ(Yn ) ∼ (13) α+1/2 , if α > −1/2 n (see Szeg˝o [9], the proof of Theorem 14.4). We prove that  log n, if α = −1/2, λ(Xmn ) ∼ |α+1/2| , if α = −1/2 . n

(14)

Here the upper estimate follows from the relations (see [9], Theorem 7.32.2 and the remarks following it)  ωn [−1,1] n−α−1/2 , if − 1 < α ≤ −1/2, ∼ (15) |ωn (±1)| 1, if α > −1/2 6

by applying Theorem 1. In case α ≥ −1/2, the lower estimate in (14) follows from Proposition 1 and (13), since 1 − y1 = 1 + yn ∼ n−2 and λ(Yn ) ∼ λ(Yn , 1). When −1 < α < −1/2, we obtain     ωn (1) ,  λ(Xmn ) ≥ λ(Xmn , 1) ≥   p (xj )ωn (y1 )(ξ − xj )  where ξ ∈ (−1, 1) is such that pm (ξ) = 1, y1 is the largest root of the Jacobi polynomial, and xj is the nearest element of Xmn to ξ such that pm (xj ) = y1 . Here ωn (1) ∼ n−1/2 , ωn (y1 ) ∼ nα+2 and

n−2 ∼ 1 − y1 ≥ C(ξ − xj )2 ≥ C|pm (xj )| · |ξ − xj |

(see [9], Theorem 7.32.4). Thus λ(Xmn ) ≥ C

n−1/2 = Cn−α−1/2 . nα+2 n−2

Summarizing, we can see by (8) and (11) that the inverse polynomial images of systems of nodes may preserve the order of the Lebesgue functions and Lebesgue constants when the systems of intervals Js possess T-polynomials. This leads to the following natural question: is the T-property necessary for the inverse polynomial image to work? The next theorem gives an affirmative answer to this question by showing that without the T condition the order of the Lebesgue constants for the inverse polynomial images of all systems of nodes is substantially larger than the optimal order log n. Theorem 2. Let Js , s ≥ 2, be given by (1) and assume that pm , m ≥ s is the Chebyshev polynomial but not a T-polynomial of degree m on Js . Then for every Yn ⊂ [−1, 1] and XN := p−1 m (Yn ) ∩ Js , n ≤ N ≤ nm, we have λ(XN ) ≥

Cn log2 n

(16)

with a constant C > 0 independent of n. Combining the above statement with Corollary 1 immediately leads to the next 7

Corollary 2. Let Js be a finite union of s ≥ 2 closed disjoint intervals on the real line, and let pm , m ≥ s, be the Chebyshev polynomial of degree m on Js . Then in order that for every Yn ⊂ (−1, 1) satisfying (10) and XN := p−1 m (Yn ) ∩ Js , n ≤ N ≤ nm we have λ(XN ) = O(λ(Yn )) it is necessary and sufficient that pm is a T-polynomial of degree m on Js . It is plausible that without the T condition the order of the Lebesgue constants for the inverse polynomial images of all systems of nodes have exponential increase, i.e., the lower estimate (16) can be essentially improved. We can verify that this is really the case for two disjoint intervals, that is when s = 2. Theorem 3. Let J2 = [−1, a] ∪ [b, 1] and assume that pm , m ≥ 3, is the Chebyshev polynomial of degree m on J2 which is not a T-polynomial on Js . Then for every Yn ⊂ [−1, 1] and XN := p−1 m (Yn ) ∩ J2 , n ≤ N ≤ nm we have λ(XN ) ≥ eCn

(17)

with a constant C > 0 independent of n. Even though the norm of Lagrange operator of degree n is of order log n, it is well known that the situation changes dramatically when the strict condition on the number of nodes of interpolation is loosened. Namely, for any ε > 0 with a proper choice of nodes in [−1, 1] one can construct polynomials of degree n which interpolate at [n(1 − ε)] points and approximate all f ∈ C[−1, 1] with optimal order En (f, [−1, 1]) where as usual En (f, K) := inf f − qn K . qn ∈Πn

In the paper [1] Erd˝os, Kro´o and Szabados gave a complete characterization of systems of nodes with the above properties on the interval [−1, 1]. It turns out that the inverse polynomial image method allows to construct such interpolating processes on any systems of disjoint intervals, as well. Theorem 4. Let s ≥ 2 and Js given by (1) be any system of disjoint intervals, and let 0 < ε < 1. Then there exist systems of nodes Xn = 8

[n(1−ε)]

{xk }k=1 ⊂ Js and mappings Qn : C(Js ) → Πn , n ∈ N such that for any f ∈ C(Js ) Qn (f, xk ) = f (xk ), 1 ≤ k ≤ [n(1 − ε)] , (18) and f − Qn (f )Js ≤ CEn (f, Js ),

(19)

where C > 0 depends only on Js and ε.

3

Proofs

In order to prove Theorem 1, we need a lemma relating the fundamental polynomials of Lagrange interpolation under the inverse polynomial images conducted using T-polynomials. First of all we rename the set Xmn in (7). Let yk = pm (xkj ),

k = 1, . . . , n; j = 1, . . . , m .

m Evidently, Xmn = {xkj }n, k=1, j=1 . Set

Ωmn (x) := ωn (pm (x)) = C

m n  

(x − xkj ).

k=1 j=1

Let k (Yn , y) :=

ωn (y)  ωn (yk )(y −

yk )

,

k = 1, . . . , n

and kj (Xmn , x) =

Ωmn (x) , Ωmn (xkj )(x − xkj )

k = 1, . . . , n; j = 1, . . . , m

be the fundamental functions of interpolation for the sets Yn and XN , respectively. Lemma 1. Let Js given by (1) possess a T-polynomial pm of degree m with 2 ≤ s ≤ m. Then for any system of nodes (6) and Xmn = p−1 m (Yn ) we have 



|kj (Xmn , x)| ≤ Cm |k (Yn , pm (x))| + |ωn (pm (x))| k = 1, . . . , n; j = 1, . . . , m; x ∈ Js . 9

  |k (Yn , 1)|  k (Yn , −1)  + , ωn (1) ωn (−1) 

Proof of Lemma 1. In what follows, c1 , c2 , . . . will denote positive constants depending on m. Let x ∈ Js , and fix k and j. We distinguish two cases. Case 1: |x − xkj | ≤ |pm (xkj )|. Then we have with y := pm (x), kj (Xmn , x) =

1 ωn (y) ·  .  ωn (yk ) pm (xkj )(x − xkj )

(20)

By Taylor expansion |y − yk | =|pm (x) − pm (xkj )| 1 ≤|pm (xkj )| · |x − xkj | + ||pm ||[−1,1] (x − xkj )2 2  ≤c1 |pm (xkj )| · |x − xkj | . Hence

    ωn (y)  = c1 |k (Yn , y)| .  |kj (Xmn , x)| ≤ c1   ω (yk )(y − yk )  n

Case 2: |x − xkj | >

|pm (xkj )|.

Then    ωn (y)  1 · |kj (Xmn , x)| ≤   .   ωn (yk ) pm (xkj )2

Let ξkj ∈ int Js , be the nearest point to xkj where |pm (ξkj )| = 1 and pm (ξkj ) = 0. Denote by Ikj ⊂ Js the interval around ξkj such that |pm (x)| ≥ c3 > 0 for x ∈ Ikj . (Such interval exists since pm (ξkj ) = 0.) Now if xkj ∈ / Ikj , then  |pm (xkj )| ≥ c4 > 0, and we get |kj (Xmn , x)| ≤

1 |ωn (y)| 2 ≤ 2 |k (Yn , y)| . 2  c4 |ωn (yk )| c4

Finally, let xkj ∈ Ikj . Without loss of generality we may assume that pm (ξkj ) = −1. We obtain yk + 1 = pm (xkj ) − pm (ξkj ) ≤ c5 (ξkj − xkj )2 . Thus c3 |pm (xkj )| = |pm (xkj ) − pm (ξkj )| = |pm (ζ)| · |xkj − ξkj | ≥ √ 1 + yk , c5 10

ζ ∈ (ξkj , xkj ) ⊂ Ikj . Hence |kj (Xmn , x)| ≤ c6 m

|ωn (y)|  |ωn (yk )|(1 +

yk )

≤C

|ωn (y)| |k (Yn , −1)| . |ωn (−1)|

Assuming pm (xkj ) = 1 we end up with the other term in the estimate.  Now the proof of Theorem 1 in case s < m easily follows simply by adding the inequalities of Lemma 1. In the special case m = s the derivative of the T-polynomial pm (x) does not vanish on Js , i.e., we have |p (x)| ≥ c > 0 on Js . Clearly, this yields that in the proof of Lemma 1 we have the same estimate in Case 2 as in Case 1, i.e., the second term in the upper bound of Lemma 1 can be omitted. Thus the proof of Theorem 1 is completed.  Proof of Proposition 1. Since s < m the T -polynomial pm has extremal points inside Js . Assume that pm (ξ) = 1 with some ξ ∈ IntIr = (ar , br ), 1 ≤ r ≤ s. Let us show first that C λ(Xmn ) ≥ √ . n 1 − y1

(21)

Clearly there exist adjacent nodes x1,j−1 , x1j ∈ Xmn such that x1,j−1 < ξ < x1j , pm (x1,j−1 ) = pm (xj ) = y1 and [x1,j−1 , x1j ] ⊂ Int Ir . Moreover, assuming that y1 > 0 (otherwise (21) is trivial), it is easy to see that x1,j−1 , x1j are separated from the endpoints of Ir by constants depending only on m and Ir . Hence using the Bernstein inequality for polynomial 1j on Ir yields 1j (Xmn , x1j ) − 1j (Xmn , x1,j−1 ) 1 = x1j − x1,j−1 x1j − x1,j−1 ≤ 1j (Xmn )[x1,j−1 ,x1j ] ≤ cn1j (Xmn )Ir ≤ cnλ(Xmn ) .

(22)

On other other hand, since pm (ξ) = 0, pm (ξ) = 0 it follows that 1 − y1 = pm (ξ) − pm (x1j ) ≥ c1 (ξ − x1j )2 , 1 − y1 ≥ c1 (x1,j−1 − ξ)2 , (23) whence x1j − x1,j−1 ≤ c 1 − y1 . This combined with (22) yields (21). Now we will show that λ(Xmn ) ≥ Cλ(Yn , 1). Denote η, ξ < η ≤ br , the nearest point to ξ such that pm (η) = −1. Then, because of the monotonicity of pm (x) in the interval [ξ, η], there exists a unique j  (depending on k) such that xkj  ∈ (ξ, η). By (20) we obtain kj  (Xmn , ξ) =

ωn (1)   ωn (yk )pm (xkj  )(ξ 11

− xkj  )

,

k = 1, . . . , n .

Here, similarly to (23) we have |pm (xkj  )| ≤ c|ξ − xkj  | ≤ c

1 − yk ,

k = 1, . . . , n,

and thus |kj  (Xmn , ξ)| ≥ c

ωn (1)  |ωn (yk )|(1 −

yk )

= c|k (Yn , 1)|,

k = 1, . . . , n .

Summing up for k = 1, . . . , n we obtain the desired inequality λ(Xmn ) ≥ Cλ(Yn , 1). If pm (ξ) = −1 with some ξ ∈ IntIr , 1 ≤ r ≤ s, then a similar argument yields lower bounds with 1 + yn and λ(Yn , −1).  Now we turn to the proof of Theorem 2. First we need the next known auxiliary statement on the so called ”needle” polynomials which is a particular case of Lemma 3 and Corollary 2 from [4]. Lemma 2. For any [a, b] ⊂ [−1, 0], n ∈ N there exist polynomials gn of degree n such that |gn | ≤ 1 on [−1, 1] \ [a, b] and c2 n(b − a) gn [a,b] ≥ c1 exp √ 1+b

(24)

with some absolute positive constants c1 , c2 . The above lemma can be extended to compact sets. Lemma 3. Let E ⊂ R be a compact set such that [−1, 1] ⊂ E and (−t, −1)∩ E = ∅ for some t > 1. Then for any [a, b] ⊂ [−1, 0], n ∈ N there exist polynomials gn ∈ Πn such that |gn | ≤ 1 on E \ [a, b] and with some c1 , c2 > 0 depending only on E c2 n(b − a) gn [a,b] ≥ c1 exp √ . 1+b

(25)

Proof of Lemma 3. Let E ⊂ [−r, r] with some r > 0. Consider a polynomial qn ∈ Πn satisfying the conditions of Lemma 2, i.e., |qn | ≤ 1 on [−1, 1] \ [a, b] and for some x0 ∈ [a, b] c2 n(b − a) |qn (x0 )| ≥ c1 exp √ . 1+b 12

(26)

Note that since [a, b] ⊂ [−1, 0] we have |qn | ≤ 1 on [0, 1] which easily yields that for some M > 0 (depending only on r) |qn | ≤ M n on E. Now for an integer d to be specified below, consider the polynomial 

(x − x0 )2 gn (x) := qn (x) 1 − (r + 2)2

dn ∈ Πn(2d+1) .

Clearly, |x − x0 | ≤ r + 1 whenever x ∈ E. Thus |gn (x)| ≤ |qn (x)| ≤ 1,

x ∈ [−1, 1] \ [a, b].

Furthermore, for x ∈ E \ [−1, 1] ⊂ E \ (−t, 1] and x0 ∈ [a, b] ⊂ [−1, 0] we have that |x − x0 | ≥ min{1, t − 1} := t0 > 0 and hence  |gn (x)| ≤ M

n

t20 1− (r + 2)2

dn ,

x ∈ E \ [−1, 1].

Evidently, the integer d can be chosen sufficiently large (and depending only on t0 , r, M ) so that the right hand side of the above estimate is at most 1. The above estimates mean that |gn | ≤ 1 whenever x ∈ E\[a, b]. Moreover, since gn (x0 ) = qn (x0 ) relation (25) follows immediately from (26).  Proof of Theorem 2. Since the Lagrange interpolation operator for mn nodes reproduces polynomials of degree at most mn − 1, we have for any polynomial gn ∈ Πmn−1 gn E ≤ λ(XN )gn XN .

(27)

Denote by γn the maximal length of a subinterval in Js free of the nodes of XN . Then Lemma 3 immediately yields that λ(XN ) ≥ c1 ec2 nγn . Hence the statement of the theorem follows trivially if γn ≥ C

log n with a n

proper C > 0. Therefore we can assume that γn = o(1). Since pm is not a T-polynomial it follows that for some subinterval [aj , bj ] ⊂ Js we have min{|pm (aj )|, |pm (bj )|} < 1. Without loss of generality [aj , bj ] = 13

[−1, 1] (this can be achieved by a linear transformation), and pm (−1) = 1 − t, 0 < t < 2. Let −1 ≤ x1 < x2 ≤ 1 be the first two points from XN in [aj , bj ] = [−1, 1]. Set δn := x2 −x1 . By the above observation 1+x2 = o(1) and hence x2 < 0 for n sufficiently large. In addition, pm (x1 ), pm (x2 ) = pm (−1)+o(1) = 1−t+o(1). Case A: 1 + x1 < x2 − x1 = δn . Then clearly 1 + x2 < 2δn . We shall apply Lemma 2 with [a, b] = [x1 , x2 ]. There exist polynomials gn of degree n − 1 such that |gn | ≤ 1 on Js \ (x1 , x2 ) and with some c1 , c2 > 0 depending only on Js c2 n(x2 − x1 ) gn [x1 ,x2 ] ≥ c1 exp √ . (28) 1 + x2 Since √ δn x2 − x1 √ > 2 1 + x2 we obtain from (28) gn E ≥ c1 ec3 n

√

δn

.

In addition, using that |gn | ≤ 1 on Js \ (x1 , x2 ) we have gn XN ≤ 1. Combining the last two bounds together with (27)) yields λ(XN ) ≥ c1 ec3 n

√

δn

.

(29)

Case B: 1+x1 ≥ δn . Lemma 2 will be used for the interval [a, b] = [−1, x1 ] yielding again √ √ qn E ≥ c1 ec2 n x1 +1 ≥ c1 ec2 n δn . (30) Hence analogously to Case A we obtain again estimate (29). Recalling that m ≥ s it follows that there exist 2 consecutive oscillation points zj , zj+1 satisfying (5) which belong to a single subinterval [ai , bi ] ⊂ Js . Since pm (xk ) = 1 − t + o(1), k = 1, 2 and 1 + x2 = o(1) it can be easily seen that (zj , zj+1 ) ∩ (x1 , x2 ) = ∅. Moreover, since pm (−1) = 1 − t, 0 < t < 2 there exist points s1 , s2 ∈ [zj , zj+1 ], s1 = s2 such that pm (sk ) = pm (xk ) = pm (−1) + o(1) = 1 − t + o(1), k = 1, 2. Since |pm (zj )| = 1 for all j-s we must have s1 , s2 ∈ (zi + ε, zi+1 − ε) ⊂ (ai + ε, bi − ε) with ε > 0 depending only on pm (and thus independent of n). 14

Using that XN = p−1 m (Yn ) ∩ Js and pm (sk ) = pm (xk ) with xk ∈ XN , k = 1, 2 we obviously have s1 , s2 ∈ XN . Furthermore, since pm = 0 in (zi , zi+1 ) it follows that |pm | ≥ c0 in (zi + ε, zi+1 − ε) with some c0 > 0 independent of n. Hence s2 − s1 ≤

1 1 |pm (s1 ) − pm (s2 )| = |pm (x1 ) − pm (x2 )| ≤ c1 (x2 − x1 ) = c1 δn , c0 c0

i.e., s2 − s1 = O(δn ) for the given s1 , s2 ∈ XN . Now consider a polynomial gn ∈ Πmn such that gn (si ) = (−1)i , i = 1, 2 and gn = 0 on all other nodes of XN . Then clearly gn Js ≤ λ(XN ). On the other hand using the classical Bernstein inequality on (ai + ε, bi − ε) 

2 = |gn (s1 ) − gn (s2 )| ≤ |s2 − s1 |gn (ai +ε,bi −ε) ≤

N |s2 − s1 |gn (ai ,bi ) ≤ c1 nδn λ(XN ), ε

i.e., λ(XN ) ≥

c2 . nδn

(31)

Finally, combining estimates (29) and (31) yields 

1 cn√δn c1 n ,e . λ(XN ) ≥ c max ≥ nδn log2 n  To prove Theorem 3 we need another lemma. For an arbitrary set of nodes XN and interval I ⊂ Js , denote ν(XN , I) := card(XN ∩ I) . Lemma 4. Let XN = {xk }N k=1 ⊂ Js ⊂ [−1, 1] be an arbitrary system of nodes. Then there exist c1 , c2 > 0 depending only on I such that whenever ν(XN , I) ≤ c1 N , then λ(XN ) ≥ ec2 N . Proof of Lemma 4. Let I1 ⊂ I be the interval with the same midpoint as I, and |I| = 2|I1 |. Then by [4] Lemma 3 there exists p ∈ Π[N/2] such that p(x) ≥ ec2 N if x ∈ I1 , and |p(x)| ≤ 1/2 if x ∈ [−1, 1] \ I , 15

where c2 > 0 depends only on I. Let QN = T[N/2] p ∈ ΠN , where Tk ∈ Πk stands for the classical k-th degree norm 1 Chebyshev polynomial on [−1, 1]. Denote by μN (I1 ) the number of oscillation points of T[N/2] in I1 . Clearly, there exists a constant c1 > 0 depending only on I such that μN (I1 ) ≥ c1 N + 3. Let q ∈ ΠN −1 be such that q(xi ) = (−1)i , i = 1, . . . , N . Then evidently, qJs ≤ λ(XN ). Now assume that νN (I) ≤ c1 N but contrary to the statement of the lemma, we have λ(XN ) < ec2 N . On I1 , at the oscillation points of TN/2 we have |QN | = p ≥ ec2 N > λ(XN ) ≥ qJs , i.e. q − QN has at least μN (I1 ) − 1 zeros in I1 . Furthermore for xi ∈ Js \ I |QN (xi )| = |TN/2 (xi )| · |p(xi )| ≤ 1/2 < |q(xi )|, i.e. q − QN has at least N − νN (I) − 1 zeros in [−1, 1] \ I. Thus the total number of zeros of polynomial q − QN ∈ ΠN is at least N − νN (I) − 1 + μN (I1 ) − 1 ≥ N − c1 N + μN (I1 ) − 2 ≥ N + 1, a contradiction.  Proof of Theorem 3. Since pm is not a T-polynomial of degree m for J2 , the set p−1 m ([−1, 1]) \ J2 must contain an interval I ⊂ [a, b]. There are at least two intervals I1 , I2 in J2 such that pm (I1 ) = pm (I2 ) = pm (I). At least one of them, say I1 , is such that I1 ⊂ (−1, 1). Evidently, ν(XN , I1 ) = ν(XN , I). If ν(XN , I) = ν(XN , I1 ) ≤ c1 n1 , then by Lemma 4 we are done. So from now on we may assume that ν(XN , I) > c1 N ≥ c1 n. Let ηn ∈ [−1, 1] be such that |ωn (ηn )| = ωn [−1,1] . The equation pm (x) = ηn has at least two solutions in J2 when m ≥ 3. Moreover, since m ≥ 3 it is easy to see that at least one of these solutions must be separated away from the endpoints ±1. Thus without loss of generality we may assume that for a solution x = ξn we have −1 + ε ≤ ξn ≤ a with some ε > 0 independent of n. If there is no element xk ∈ XN with −1 ≤ xk ≤ −1 + ε/2, then λ(XN ) is of exponential order.  Otherwise, with the notation q(x) = (x − xj ) we have xj ∈I

   q(xk )  νN (I)   ≥ e c2 n  q(ξn )  ≥ c(ε) 16

where c(ε) > 1 is independent of n. ωn (pm (x)) Let ΩN (x) = , then q(x) ΩN (xk ) =

pm (xk )ωn (pm (xk )) . q(xk )

Using the last estimate and Markov’s inequality for the polynomial ωn we have,      q(xk )    (η ) ω n n ·  |k (ξn )| =      q(ξn ) pm (xk )ωn (pm (xk ))(ξn − xk )  ωn [−1,1] c e c2 n ≥ 2 e c2 n ≥ e c 3 n , ≥   2pm [−1,1] ωn [−1,1] n whence the statement of the theorem.  Theorem 1 gives the possibility of constructing sets of nodes of interpolation with ”good” Lebesgue constant. The drawback of the construction is that it provides a subsequence of systems of nodes. Nevertheless, such a subsystem can be easily extended for all n’s, using the following lemma. Lemma 5. Let (2) be an arbitrary system of nodes in Js , let ωn (x) =

n 

(x − xk ),

k=1

and let xn+1 ∈ Js be such that |ωn (xn+1 )| = ωn Js . Denoting Xn+1 = Xn ∪ {xn+1 }, we have λ(Xn+1 ) ≤ 2λ(Xn ) + 1 . Proof. We have

   n      ωn (x)  (x − xn+1 )ωn (x)     λ(Xn+1 , x) =  (xk − xn+1 )ω  (xk )(x − xk )  +  ωn (xn+1 )  n k=1    n       ωn (x) ωn (x)     ≤ +  (x − xk )ω  (xk )   (xn+1 − xk )ω  (xk )  + 1 n n k=1 ≤λ(Xn ) + λ(Xn , xn+1 ) + 1 ≤2λ(Xn ) + 1 .

17

Corollary 3. Let {nk }∞ k=1 be an increasing sequence of positive integers such that nk+1 − nk ≤ q, k = 1, 2, . . . , where q is a positive integer. Further let {Xnk }∞ k=1 ⊂ Js be a sequence of nodes of interpolation. Then there exists a sequence of nodes {Xn }∞ n=1 ⊂ Js such that Xn = Xnk if n = nk , and λ(Xn ) ≤ 3q λ(Xnk )

f or all

nk ≤ n < nk+1 .

Corollary 3 can be easily obtained by iterating Lemma 5. The proof of Theorem 4 is based on the following lemma which is a modification of a construction developed in [1]. Let Xp = {xk }pk=1 and Yq = {yj }qj=1 be arbitrary sets of nodes of interpolation in Js and let Ij (ε) := {xk : yj ≤ xk < yj+1 , yj ∈ Js (ε)},

j = 1, . . . , q − 1 ,

where Js (ε) :=

s 

[ai + ε, bi − ε],

0 < ε < ε0 :=

i=1

Furthermore, set γ(Xp , Yq , ε) := max

1≤j≤q−1

1 min (bi − ai ) . 2 1≤i≤s



1.

xk ∈Ij (ε)

For an arbitrary f ∈ C(Js ), s ≥ 2, consider the linear operator q−1    j (Yq , x) + j+1 (Yq , x) 2 k (Xp , x)f (xk ) . Sp,q (f, x) := j (Yq , xk ) + j+1 (Yq , xk ) j=1 xk ∈Ij (ε)

Lemma 6. For any 0 < ε < ε0 and f ∈ C(Js ), (i) Sp,q (f ) is a polynomial of degree at most p + 2q − 3, (ii) Sp,q (f, xk ) = f (xk ) for xk ∈ Js (ε), (iii) and we have Sp,q (f )Js

q  ≤ Cf Js γ(Xp , Yq , ε) j (Yq )2 j=1

18

max k (Xp )Js .

1≤k≤p Js

(32)

Proof. (i)-(ii) can be seen directly from the definition of the operator and the fundamental functions. (iii) According to a result of Erd˝os and Tur´an [2], Lemma IV, we have j (Yq , y) + j+1 (Yq , y) ≥ 1,

yj ≤ y ≤ yj+1 , 1 ≤ j ≤ q − 1 .

This combined with the inequality q−1 

2

(j (Yq , x) + j+1 (Yq , x)) ≤ 4

j=1

q 

j (Yq , x)2 ,

x ∈ [−1, 1]

j=1

yields the third statement of the lemma.  Proof of Theorem 4. Let 0 < ε < ε0 where ε0 is defined in (32). According to Totik [11], Theorem 2 there exists Js (ε)  Js such that |Js (ε) \ Js | < ε2

(33)

and Js (ε) possesses a T-polynomial pm of degree m = m(ε). (Note that if Js itself has a T-polynomial, then m is independent of ε, and we take Js (ε) = Js .) We will apply Lemma 6 with ε2 instead of ε and with     2   n(1 − 2ε2 ) + 1 nε + 1 p= + 1 m and q = − 1 m. m m Denote by Zr the set of roots of the Chebyshev polynomial of degree r on [−1, 1], and let Xp := p−1 m (Zp/m ),

Yq := p−1 m (Zq/m ) ⊂ Js (ε).

Then by Lemma 6(i) the degree of the operator Sp,q will be p + 2q − 3 ≤ n(1 − 2ε2 ) + 2nε2 = n . Let now Pn (f ) be the best approximating polynomial of f ∈ C(Js ) of degree at most n, and define Qn (f, x) := Pn (f, x) + Sp,q (f − Pn (f ), x) . This is a polynomial of degree at most n, and by Lemma 6(ii), it interpolates f at the points xk ∈ Js (ε2 ). The number of xk ’s in Js (ε) is p, but the operator 19

does not interpolate at xk ∈ Js (ε) \ Js (ε2 ). According to (33), the measure of the latter set does not exceed ε2 + 2sε2 = (1 + 2s)ε2 . Apparently, the image pm (Js (ε) \ Js (ε2 )) is of the form I := [−1, −1 + α] ∪ [1 − β, 1], where max(α, β) ≤ c1 ε2 . The number of xk ’s in Js (ε) \ Js (ε2 ) is the same as the number of Chebyshev nodes from Zp/m in I, i.e. ≤ c2 nε/m ≤ c2 nε. Thus the number of interpolation points is ≥ p − c2 nε ≥ n(1 − c3 ε) which, after modifying the ε with a proper multiplicative constant, is the same as stated in the theorem. Finally, we prove the estimate for the rate of convergence. By Lemma 6(iii), ⎞ ⎛ q  f −Qn (f )Js ≤ En (f, Js ) ⎝1 + γ(Xp , Yq , ε2 ) j (Yq )2 max k (Xp )Js ⎠ . 1≤k≤p j=1

Js

The number of xk ’s in each interval Ij (ε2 ) is the same as the number of Chebyshev nodes in the corresponding interval, and this number is the ratio of the degrees of the the two sets of Chebyshev nodes, i.e. γ(Xp , Yq , ε2 ) ≤

c . ε2

On the other hand, by Lemma 1 each fundamental function of interpolation based on Xp or Yq can be estimated on Js (ε) by some fundamental function of interpolation based on roots of some Zr , at the corresponding argument in [−1, 1]. Since the sum of squares of fundamental functions of interpolation based on Zr are bounded by 2 (cf. Fej´er [3], p. 5), collecting our estimates, we get the statement of the theorem.  Acknowledgement. The authors are grateful to the anonymous referee for the thorough reading of the original manuscript and for numerous helpful suggestions.

References [1] P. Erd˝os, A. Kro´o and J. Szabados, On convergent interpolatory polynomials, J. Approx. Theory, 58(1989), 232-241. [2] P. Erd˝os and P. Tur´an, On interpolation III, Ann. of Math., 41 (1940), 510-553. 20

[3] L. Fej´er, Lagrangesche Interpolation und die zugeh¨origen konjugierten Punkte, Math. Annalen, 106 (1932), 1-55. [4] A. Kro´o and J. Swetits, On density of interpolation points, a Kadec-type theorem, and Saff’s principle of contamination in Lp -approximation, Constr. Approx., 8 (1992), 87-103. [5] A. L. Lukashov and J. Szabados, The order of Lebesgue constant of Lagrange interpolation on several intervals, Periodica Math. Hungar., to appear. [6] F. Peherstorfer, Orthogonal and Chebyshev polynomials on two intervals, Acta Math. Hungar., 55 (1990), 245-278. [7] F. Peherstorfer, Deformation of minimizing polynomials and approximation of several intervals by an inverse polynomial mapping, J. Approx. Th., 111 (2001), 180-195. [8] F. Peherstorfer and K. Schiefermayr, Description of extremal polynomials on several intervals and their computation.I, Acta Math. Hungar., 83 (1999), 27-58. [9] G. Szeg˝o, Orthogonal Polynomials, AMS Publications No. 23 (Providence, 1939). [10] V. Totik, Polynomial inverse images and polynomial inequalities, Acta Math., 187 (2001), 139-160. [11] V. Totik, Chebyshev constants and the inheritance problem, J. Approx. Th., 160 (2009), 187-201.

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